Band gaps of the phononic piezoelectric smart materials with LCR shunting circuits

Environmental forces can produce undesired vibrations in mechanical structures that can limit the precision of mechanical equipment and cause mechanical failure. Piezoelectric-shunt damping is an attractive technique for controlling the vibrating structures, which is reliable, economical and light-weight. Phononic crystal is an internal component whose elastic constant, density and sound velocity change periodically. When the elastic wave passes through a phononic crystal, special dispersion curve is formed due to the interaction of periodic arrangement materials. In order to study the electromagnetic oscillation band gap of the piezoelectric phononic crystal with LCR shunt network at torsional and flexural vibration, we propose a new phononic piezoelectric beam, which is composed of aluminum and epoxy resin. When the piezoelectric patch is strained, the electrical energy is dissipated as current flows through an external LCR shunting circuit. By combining the piezoelectric effect with the mechanical vibration of the smart material, the equivalent additional stress of piezoelectric patches is deduced. Moreover, coupling the energy band theory of phononic crystal with the effect of electromagnetic oscillation, we calculate the band gap characteristics of torsional and flexural vibration of intelligent material. Using the transfer matrix method and Bloch theorem for periodic boundary conditions, the band gap of the phononic beam can be calculated. With the increase of resistance, the amplitude attenuation of the band gap decreases. However, it can expand the frequency range. The inherent frequency of the electromagnetic oscillation is 1/[2π√L(C + CP)]. The sum of capacitance and inherent capacitance is the total capacitance of the shunting circuit. Therefore, the frequency of the electromagnetic oscillation decreases with the increases of the capacitance and inductance. The amplitude attenuation of the band gap increases with the increase of the inductance and decreases greatly with the rise of the capacitance. Three main differences between the LCR shunt networks and traditional circuits are found. First, the band gaps of the phononic piezoelectric smart material are composed of Bragg band gaps and local resonant band gaps. The former one is due to the mismatch between aluminum and epoxy resin, which makes the elastic waves have no corresponding vibration modes at certain frequencies. The latter one is from the effect of electromagnetic oscillation in LCR shunt networks, which consume the energy by resistor. Second, by tuning the resistance, capacitance and inductance, we can change the singularity position and stress magnitude of equivalent additional force curve. The amplitude attenuations of locally resonant band gaps and electromechanical coupling coefficient will be changed. Third, both locations and widths of the band gaps can be tuned by simply varying the value of negative capacitance of the shunting networks without needing to modify the configuration of the structure. Therefore, it provides a new idea for controlling the vibration and reducing the noise of the phononic piezoelectric smart material.